121 research outputs found
Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space
A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time ?^*(2^{?(twlog tw)}). Using their inspired Cut&Count technique, they obtained ?^*(?^tw) time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely believed to be unavoidable. In contrast, for the slightly larger parameter called treedepth, there are already several examples of matching the time bounds obtained for treewidth, but using only polynomial space. Nevertheless, this has remained open for connectivity problems.
In the present work, we close this knowledge gap by applying the Cut&Count technique to graphs of small treedepth. While the general idea is unchanged, we have to design novel procedures for counting consistently cut solution candidates using only polynomial space. Concretely, we obtain time ?^*(3^d) and polynomial space for Connected Vertex Cover, Feedback Vertex Set, and Steiner Tree on graphs of treedepth d. Similarly, we obtain time ?^*(4^d) and polynomial space for Connected Dominating Set and Connected Odd Cycle Transversal
Dense peelable random uniform hypergraphs
We describe a new family of k-uniform hypergraphs with independent random edges. The hypergraphs have a high probability of being peelable, i.e. to admit no sub-hypergraph of minimum degree 2, even when the edge density (number of edges over vertices) is close to 1.
In our construction, the vertex set is partitioned into linearly arranged segments and each edge is incident to random vertices of k consecutive segments. Quite surprisingly, the linear geometry allows our graphs to be peeled "from the outside in". The density thresholds f_k for peelability of our hypergraphs (f_3 ~ 0.918, f_4 ~ 0.977, f_5 ~ 0.992, ...) are well beyond the corresponding thresholds (c_3 ~ 0.818, c_4 ~ 0.772, c_5 ~ 0.702, ...) of standard k-uniform random hypergraphs.
To get a grip on f_k, we analyse an idealised peeling process on the random weak limit of our hypergraph family. The process can be described in terms of an operator on [0,1]^Z and f_k can be linked to thresholds relating to the operator. These thresholds are then tractable with numerical methods.
Random hypergraphs underlie the construction of various data structures based on hashing, for instance invertible Bloom filters, perfect hash functions, retrieval data structures, error correcting codes and cuckoo hash tables, where inputs are mapped to edges using hash functions. Frequently, the data structures rely on peelability of the hypergraph or peelability allows for simple linear time algorithms. Memory efficiency is closely tied to edge density while worst and average case query times are tied to maximum and average edge size.
To demonstrate the usefulness of our construction, we used our 3-uniform hypergraphs as a drop-in replacement for the standard 3-uniform hypergraphs in a retrieval data structure by Botelho et al. [Fabiano Cupertino Botelho et al., 2013]. This reduces memory usage from 1.23m bits to 1.12m bits (m being the input size) with almost no change in running time. Using k > 3 attains, at small sacrifices in running time, further improvements to memory usage
Repetition Detection in a Dynamic String
A string UU for a non-empty string U is called a square. Squares have been well-studied both from a combinatorial and an algorithmic perspective. In this paper, we are the first to consider the problem of maintaining a representation of the squares in a dynamic string S of length at most n. We present an algorithm that updates this representation in n^o(1) time. This representation allows us to report a longest square-substring of S in O(1) time and all square-substrings of S in O(output) time. We achieve this by introducing a novel tool - maintaining prefix-suffix matches of two dynamic strings.
We extend the above result to address the problem of maintaining a representation of all runs (maximal repetitions) of the string. Runs are known to capture the periodic structure of a string, and, as an application, we show that our representation of runs allows us to efficiently answer periodicity queries for substrings of a dynamic string. These queries have proven useful in static pattern matching problems and our techniques have the potential of offering solutions to these problems in a dynamic text setting
Resilient Level Ancestor, Bottleneck, and Lowest Common Ancestor Queries in Dynamic Trees
We study the problem of designing a resilient data structure maintaining a tree under the Faulty-RAM model [Finocchi and Italiano, STOC\u2704] in which up to ? memory words can be corrupted by an adversary. Our data structure stores a rooted dynamic tree that can be updated via the addition of new leaves, requires linear size, and supports resilient (weighted) level ancestor queries, lowest common ancestor queries, and bottleneck vertex queries in O(?) worst-case time per operation
On the Complexity of BWT-Runs Minimization via Alphabet Reordering
The Burrows-Wheeler Transform (BWT) has been an essential tool in text
compression and indexing. First introduced in 1994, it went on to provide the
backbone for the first encoding of the classic suffix tree data structure in
space close to the entropy-based lower bound. Recently, there has been the
development of compact suffix trees in space proportional to "", the number
of runs in the BWT, as well as the appearance of in the time complexity of
new algorithms. Unlike other popular measures of compression, the parameter
is sensitive to the lexicographic ordering given to the text's alphabet.
Despite several past attempts to exploit this, a provably efficient algorithm
for finding, or approximating, an alphabet ordering which minimizes has
been open for years.
We present the first set of results on the computational complexity of
minimizing BWT-runs via alphabet reordering. We prove that the decision version
of this problem is NP-complete and cannot be solved in time unless the Exponential Time Hypothesis fails, where is the
size of the alphabet and is the length of the text. We also show that the
optimization problem is APX-hard. In doing so, we relate two previously
disparate topics: the optimal traveling salesperson path and the number of runs
in the BWT of a text, providing a surprising connection between problems on
graphs and text compression. Also, by relating recent results in the field of
dictionary compression, we illustrate that an arbitrary alphabet ordering
provides a -approximation.
We provide an optimal linear-time algorithm for the problem of finding a run
minimizing ordering on a subset of symbols (occurring only once) under ordering
constraints, and prove a generalization of this problem to a class of graphs
with BWT like properties called Wheeler graphs is NP-complete
On Computing Homological Hitting Sets
Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. In this paper, we initiate the algorithmic study of a high-dimensional cut problem. The problem we study, namely, Homological Hitting Set (HHS), is defined as follows: Given a nontrivial r-cycle z in a simplicial complex, find a set ? of r-dimensional simplices of minimum cardinality so that ? meets every cycle homologous to z. Our first result is that HHS admits a polynomial-time solution on triangulations of closed surfaces. Interestingly, the minimal solution is given in terms of the cocycles of the surface. Next, we provide an example of a 2-complex for which the (unique) minimal hitting set is not a cocycle. Furthermore, for general complexes, we show that HHS is W[1]-hard with respect to the solution size p. In contrast, on the positive side, we show that HHS admits an FPT algorithm with respect to p+?, where ? is the maximum degree of the Hasse graph of the complex ?
Partitioning the Bags of a Tree Decomposition into Cliques
We consider a variant of treewidth that we call clique-partitioned treewidth in which each bag is partitioned into cliques. This is motivated by the recent development of FPT-algorithms based on similar parameters for various problems. With this paper, we take a first step towards computing clique-partitioned tree decompositions.
Our focus lies on the subproblem of computing clique partitions, i.e., for each bag of a given tree decomposition, we compute an optimal partition of the induced subgraph into cliques. The goal here is to minimize the product of the clique sizes (plus 1). We show that this problem is NP-hard. We also describe four heuristic approaches as well as an exact branch-and-bound algorithm. Our evaluation shows that the branch-and-bound solver is sufficiently efficient to serve as a good baseline. Moreover, our heuristics yield solutions close to the optimum. As a bonus, our algorithms allow us to compute first upper bounds for the clique-partitioned treewidth of real-world networks. A comparison to traditional treewidth indicates that clique-partitioned treewidth is a promising parameter for graphs with high clustering
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